First, replace the battery.... if it is even the slightest but shady go new!
Second, someone is going to tell you to head over to Motorrad Elektric
and buy Rick's book titled "Classic Boxer Charging
"... unfortunately Rick doesn't have any left and isn't planning on printing a new edition any time soon.
12v is way too low, and you do have something wrong other than the battery for absolute certain... there are a number of things to check and methods to check them... my own voyage of discover his here...
"OK charging system Guru's.. WTF is up with my GS"
I would say to start by checking the rotor.... that's my guess. Although when my former R100RS had a similar issue, it was the brushes.
My current R80G/S throws about 13.7 volts at 4K rpm's.... all on the stock charging system